^{1}and Yue-Sheng Wang

^{1,a)}

### Abstract

The bandgap properties of an open or covered phononic crystal slab with resonators are studied by using the finite element method. The results show that complete bandgap comes up for the proposed slabs with resonators due to the local resonance. The effects of the topological geometry of the resonators on the bandgaps are discussed, and optimal geometry is suggested. The mechanism of the bandgap generation is analyzed by studying the eigen modes at the bandgap edges. Equivalent spring-mass/pendulum models are developed to predict the eigen frequencies at the lower bandgap edges. The evaluated results obtained by the equivalent models are in general agreement with the numerical ones. The work in this paper is indispensable to the optimal design of the bandgaps of phononic crystal slabs.

The authors are grateful for the support from the National Eigen Science Foundation of China (Grant Nos. 51178037 and 11272041) and the National Basic Research Program of China (2010CB732104).

I. INTRODUCTION

II. MODEL AND FORMULATION

III. RESULTS AND DISCUSSIONS

A. Open PC slabs

1. Effects of the connector distribution

2. Effects of the slab thickness

3. Effects of the resonator size

B. Covered PC slabs

1. Effects of the connector distribution

2. Effects of the cover thickness

3. Effects of the connector size

4. Effects of the lump size

C. Equivalent spring-mass/pendulum models for lower bandgap edge modes

1. Open PC slabs

2. Covered PC slabs

IV. CONCLUDING REMARKS

### Key Topics

- Band gap
- 144.0
- Connectors
- 100.0
- Band structure
- 9.0
- Photonic crystals
- 8.0
- Finite element methods
- 7.0

## Figures

Unit cells and finite element models of (a) open and (b) covered PC slabs and the associated geometry parameters.

Unit cells and finite element models of (a) open and (b) covered PC slabs and the associated geometry parameters.

Cross-section of (a) the open or (b) covered PC slab parallel and (c) perpendicular with the thickness direction.

Cross-section of (a) the open or (b) covered PC slab parallel and (c) perpendicular with the thickness direction.

Cross-sections of the unit cells in the median plane and their corresponding irreducible Brillouin zone.

Cross-sections of the unit cells in the median plane and their corresponding irreducible Brillouin zone.

Band structures of various open PC slabs: (a) Slab “O1”; (b) Slab “O2(a)”; (c) Slab “O2(o)”; (d) Slab “O3”; (e) Slab “O4”. The geometry parameters are b/a = 0.9, c/a = 0.8, d/a = 0.05 and h/a = 1. For comparison, the band structure for a PC slab with cuboid holes (b/a = 0.9 and h/a = 1) is shown in panel (f).

Band structures of various open PC slabs: (a) Slab “O1”; (b) Slab “O2(a)”; (c) Slab “O2(o)”; (d) Slab “O3”; (e) Slab “O4”. The geometry parameters are b/a = 0.9, c/a = 0.8, d/a = 0.05 and h/a = 1. For comparison, the band structure for a PC slab with cuboid holes (b/a = 0.9 and h/a = 1) is shown in panel (f).

x-y cross-section of the eigen modes at the bandgap edges marked in Fig. 3

Effects of the thickness of slab “O2(a)” on the bandgaps (c/a = 0.8, d/a = 0.05). The thick solid line shows the predicted result obtained by the equivalent model. The dashed and dashed-dotted lines represent the numerical results of the lower and upper edges of the bandgap, respectively.

Effects of the thickness of slab “O2(a)” on the bandgaps (c/a = 0.8, d/a = 0.05). The thick solid line shows the predicted result obtained by the equivalent model. The dashed and dashed-dotted lines represent the numerical results of the lower and upper edges of the bandgap, respectively.

Effects of the size of the lumps c/a (with d/a = 0.05, h/a = 1) (a) or the connectors d/a (with c/a = 0.8, h/a = 1) (b) on the bandgaps of slab “O2(a).”

Effects of the size of the lumps c/a (with d/a = 0.05, h/a = 1) (a) or the connectors d/a (with c/a = 0.8, h/a = 1) (b) on the bandgaps of slab “O2(a).”

Band structures for (a) slab “C1-0” and (b) covered PC slab with cubic holes. The geometrical parameters are b/a = 0.9, c/a = 0.8, d/a = 0.05 and h/a = 1.

Band structures for (a) slab “C1-0” and (b) covered PC slab with cubic holes. The geometrical parameters are b/a = 0.9, c/a = 0.8, d/a = 0.05 and h/a = 1.

Eigen modes in x-y plane at the bandgap edges of slab “C1-0” as shown in Fig. 7(a) .

Band structures for various covered PC slabs (a) slab “C2(a)-0”; (b) slab “C2(o)-0”; (c) slab “C2(a)-1”; (d) slab “C2(a)-2”; (e) slab “C4-1”; (f) slab “C4-2”. The geometrical parameters are b/a = 0.9, c/a = 0.8, d/a = 0.05, and h/a = 1.

Band structures for various covered PC slabs (a) slab “C2(a)-0”; (b) slab “C2(o)-0”; (c) slab “C2(a)-1”; (d) slab “C2(a)-2”; (e) slab “C4-1”; (f) slab “C4-2”. The geometrical parameters are b/a = 0.9, c/a = 0.8, d/a = 0.05, and h/a = 1.

Eigen modes in x-y plane [(a)–(c)] and z-x plane [(d)–(j)] at the bandgap edges for covered PC slabs shown in Fig. 10 .

Eigen modes in x-y plane [(a)–(c)] and z-x plane [(d)–(j)] at the bandgap edges for covered PC slabs shown in Fig. 10 .

(a) Effects of the cover thickness on the bandgaps for slab “C2(a)-2” (with c/a = 0.8, d/a = 0.05, b/a = 0.9, and h/a = 1) (b) x-z cross-section of the eigen mode at the lower edge of the bandgap for slab ‘C2(a)-2’ with .

(a) Effects of the cover thickness on the bandgaps for slab “C2(a)-2” (with c/a = 0.8, d/a = 0.05, b/a = 0.9, and h/a = 1) (b) x-z cross-section of the eigen mode at the lower edge of the bandgap for slab ‘C2(a)-2’ with .

Effects of the size of the connectors d/a (with c/a = 0.8, = 0.05, b/a = 0.9, and h/a = 1) (a) or (with c/a = 0.8, d/a = 0.05, b/a = 0.9, and h/a = 1) (b) on the bandgaps for slab “C2(a)-2.”

Effects of the size of the connectors d/a (with c/a = 0.8, = 0.05, b/a = 0.9, and h/a = 1) (a) or (with c/a = 0.8, d/a = 0.05, b/a = 0.9, and h/a = 1) (b) on the bandgaps for slab “C2(a)-2.”

Effects of the size of the lump c/a (h 1/a = 0.8, d/a = 0.05, b/a = 0.9, and h/a = 1) (a) or h 1/a (c/a = 0.8, d/a = 0.05, b/a = 0.9, and h/a = 1) (b) on the bandgaps for slab “C2(a)-2.”

Effects of the size of the lump c/a (h 1/a = 0.8, d/a = 0.05, b/a = 0.9, and h/a = 1) (a) or h 1/a (c/a = 0.8, d/a = 0.05, b/a = 0.9, and h/a = 1) (b) on the bandgaps for slab “C2(a)-2.”

## Tables

Open PC slab with different connector distributions for b/a = 0.9 and h/a = 1.

Open PC slab with different connector distributions for b/a = 0.9 and h/a = 1.

Covered PC slab with different connector distributions for b/a = 0.9 and h/a = 1.

Covered PC slab with different connector distributions for b/a = 0.9 and h/a = 1.

Calculation of the effective stiffness of the proposed spring-mass/pendulum models for the lower edge modes of the open PC slab.

Calculation of the effective stiffness of the proposed spring-mass/pendulum models for the lower edge modes of the open PC slab.

Calculation of the effective stiffness of the proposed spring-mass/pendulum models for the lower edge modes of the covered PC slab.

Calculation of the effective stiffness of the proposed spring-mass/pendulum models for the lower edge modes of the covered PC slab.

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